From 1460726f63208cad4cadc44e586915776fba7451 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Fri, 23 Jul 2021 23:18:06 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E6=9B=B2=E7=BA=BF=E6=9B=B2?= =?UTF-8?q?=E9=9D=A2=E7=A7=AF=E5=88=86?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ...efinite-integral-and-definite-integral.pdf | Bin 199898 -> 204401 bytes ...efinite-integral-and-definite-integral.tex | 28 ++++++++++++++++++ ...ral-calculus-of-multivariate-functions.pdf | Bin 159831 -> 164714 bytes ...ral-calculus-of-multivariate-functions.tex | 21 +++++++++++-- 4 files changed, 47 insertions(+), 2 deletions(-) diff --git a/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf b/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf index 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\textcolor{aqua}{\textbf{定理:}}设$f(x)$在$[a,b]$上连续,函数$x=\varphi(t)$满足\ding{172}$\varphi(\alpha)=a,\varphi(\beta)=b$,\ding{173}$\varphi(t)$在$[\alpha,\beta]$上具有连续导数,且其值域$R_\varphi=[a,b]$(值域超出而其他条件满足时也成立),则有$\int_a^bf(x)\,\textrm{d}x=\int_\alpha^\beta f[\varphi(t)]\varphi'(t)\,\textrm{d}t$。 +定积分的换元和不定积分的换元方法类似,其中重点就是$a$到$\alpha$,$b$到$\beta$的区间变化,从$x$的区间到$\varphi(t)$的区间,需要代入方程使其与原区间相等,从而得到$t$的区间。 + +\textbf{例题:}计算$\int_0^\pi\sqrt{\sin^3x-\sin^5x}\,\textrm{d}x$。 + +解:由于$\sqrt{\sin^3x-\sin^5x}=\sqrt{\sin^3(1-\sin^2x)}=\sin^\frac{3}{2}x\cdot\vert\cos x\vert$。 + +注意保留绝对值,若是不保存就会发现后面计算会得到上下限为同样的值从而导致错误。 + +利用积分的区间可加性将$\vert\cos x\vert$进行拆分。 + +$=\int_0^\frac{\pi}{2}\sin^\frac{3}{2}x\cos x\,\textrm{d}x+\int_\frac{\pi}{2}^\pi\sin^\frac{3}{2}x(-\cos x)\,\textrm{d}x=\int_0^\frac{\pi}{2}\sin^\frac{3}{2}x\,\textrm{d}(\sin x)-$\\$\int_\frac{\pi}{2}^\pi\sin^\frac{3}{2}x\,\textrm{d}(\sin x)=\dfrac{2}{5}\sin^\frac{5}{2}x|_0^\frac{\pi}{2}-\dfrac{2}{5}\sin^\frac{5}{2}x|_\frac{\pi}{2}^\pi=\dfrac{4}{5}$。 + +\textcolor{aqua}{\textbf{定理:}}设$f(x)$在$[0,1]$上连续: + +(1)$\int_0^\frac{\pi}{2}f(\sin x)\,\textrm{d}x=\int_0^\frac{\pi}{2}f(\cos x)\,\textrm{d}x$。 + +(2)$\int_0^\frac{\pi}{2}xf(\sin x)\,\textrm{d}x=\dfrac{\pi}{2}\int_0^\pi f(\sin x)\,\textrm{d}x$。 + +(1)证明:这个证明式子很像华莱士公式。不过华莱士给定函数$f(x)=x^n$。 + +这个式子需要将$f(\sin x)$变为$f(\cos x)$,即使用诱导公式变化。 + +$\sin\left(\dfrac{\pi}{2}-x\right)=\cos x$,所以令$x=\dfrac{\pi}{2}-t$,$t=\dfrac{\pi}{2}-x$,当$x=0$,$t=\dfrac{\pi}{2}$,当$x=\dfrac{\pi}{2}$,$t=0$,$\textrm{d}x=-\textrm{d}t$。 + +$\int_0^\frac{\pi}{2}f(\sin x)\,\textrm{d}x=-\displaystyle{\int_\frac{\pi}{2}^0f\left(\sin\dfrac{\pi}{2}-t\right)\,\textrm{d}t}=$$-\int_\frac{\pi}{2}^0f(\cos t)\,\textrm{d}t=\int^\frac{\pi}{2}_0f(\cos x)\,\textrm{d}x$。 + +(2)证明: + \subsubsection{分部积分法} \textcolor{aqua}{\textbf{定理:}}$\int_a^bu\,\textrm{d}v=[uv]_a^b-\int_a^bv\,\textrm{d}u$。 diff --git a/advanced-math/knowledge/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.pdf 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\leavevmode \medskip -\textbf{例题:}计算$\oint_\Gamma\vert y\vert\,\textrm{d}s$,其中$Gamma$为球面$x^2+y^2+z^2=2$与平面$x=y$的交线。 +\textbf{例题:}计算$\oint\limits_\Gamma\vert y\vert\,\textrm{d}s$,其中$Gamma$为球面$x^2+y^2+z^2=2$与平面$x=y$的交线。 解:根据普通对称性,对于$\vert y\vert$而言,其他象限的函数值都与第一象限区域的函数相等。令第一象限区域为$\Gamma_1$: -$\therefore\oint_\Gamma\vert y\vert\,\textrm{d}s=4\oint_{\Gamma_1}y\,\textrm{d}s$ +$\therefore\oint\limits_\Gamma\vert y\vert\,\textrm{d}s=4\oint\limits_{\Gamma_1}y\,\textrm{d}s$ 根据交线方程进行联立,得到$x=\cos t$,$y=\cos t$,$z=\sqrt{2}\sin t$。 @@ -578,6 +578,23 @@ $\therefore8\iint\limits_{\Sigma_1}y\,\textrm{d}S=\dfrac{8}{3}\iint\limits_{\Sig 是考试的重点,基本上都会考到。非常重要。 +\textbf{例题:}计算曲线积分$\oint\limits_L\dfrac{x\textrm{d}y-y\textrm{d}x}{4^x2+y^2}$,其中$L$是以点$(1,0)$为圆心,$R\geqslant1$为半径的圆,取逆时针方向。 + +解:由于是逆时针在$L$上,所以是正向:$=\displaystyle{\oint\limits_{L^+}\left(\dfrac{-y}{4x^2+y^2}\textrm{d}x+\dfrac{x}{4x^2+y^2}\textrm{d}y\right)}$。 + +又对于$L$所围成的圆面$D$,因为$4x^2+y^2\neq0$,所以$(0,0)$应该被挖去。 + +因为逆时针的方向下挖去这个点做的运动顺时针是负方向的,所以令其为$C^-$。 + +又因为格林公式$\oint\limits_{L^++C^-}P\,\textrm{d}x+Q\,\textrm{d}y=\displaystyle{\iint\limits_D\left(\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y}\right)\textrm{d}\sigma}=$\\$\displaystyle{\iint\limits_D\left(\dfrac{4x^2+y^2-3x^2}{(4x^2+y^2)^2}-\dfrac{-(4x^2+y^2)+2y^2}{(4x^2+y^2)^2}\right)\textrm{d}\sigma}=0$。旋度为0。 + +$=\oint\limits_{L^++C^-}-\oint\limits_{C^-}=0-\oint\limits_{C^-}=\oint\limits_{C^+}$。取$C:4x^2+y^2=\delta^2$,$\delta$为一个足够小的常数。(分母取$\delta^2$) + +$=\displaystyle{\oint\limits_{C^+}\left(\dfrac{-y}{4x^2+y^2}\textrm{d}x+\dfrac{x}{4x^2+y^2}\textrm{d}y\right)}=\displaystyle{\oint\limits_{C^+}\left(\dfrac{-y}{\delta^2}\textrm{d}x+\dfrac{x}{\delta^2}\textrm{d}y\right)}$ + +$=\dfrac{1}{\delta^2}\oint\limits_{C^+}-y\,\textrm{d}x+x\,\textrm{d}y$,利用格林公式,$C^+$所成区域为$D'$:$\dfrac{1}{\delta^2}\oint\limits_{D'}(1-(-1))\,\textrm{d}\sigma=\dfrac{2}{\delta^2}D'=\dfrac{2}{\delta^2}\pi\dfrac{\delta}{2}\delta=\pi$。 + + \section{第二型曲面积分} \subsection{概念}